Question: $ D = \left[\begin{array}{rr}-1 & -1 \\ -1 & -2\end{array}\right]$ $ E = \left[\begin{array}{rrr}0 & 5 & 3 \\ 4 & 5 & 2\end{array}\right]$ What is $ D E$ ?
Because $ D$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ D E = \left[\begin{array}{rr}{-1} & {-1} \\ {-1} & {-2}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{5} & \color{#9D38BD}{3} \\ {4} & \color{#DF0030}{5} & \color{#9D38BD}{2}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{0}+{-1}\cdot{4} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{0}+{-1}\cdot{4} & ? & ? \\ {-1}\cdot{0}+{-2}\cdot{4} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{0}+{-1}\cdot{4} & {-1}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{5} & ? \\ {-1}\cdot{0}+{-2}\cdot{4} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-1}\cdot{0}+{-1}\cdot{4} & {-1}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{5} & {-1}\cdot\color{#9D38BD}{3}+{-1}\cdot\color{#9D38BD}{2} \\ {-1}\cdot{0}+{-2}\cdot{4} & {-1}\cdot\color{#DF0030}{5}+{-2}\cdot\color{#DF0030}{5} & {-1}\cdot\color{#9D38BD}{3}+{-2}\cdot\color{#9D38BD}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-4 & -10 & -5 \\ -8 & -15 & -7\end{array}\right] $